Editing Characteristic subgroup (section)

== Related concepts ==

=== Normal subgroup ===
{{main|Normal subgroup}}
A subgroup of {{math|''H''}} that is invariant under all inner automorphisms is called [[normal subgroup|normal]]; also, an invariant subgroup.
:{{math|∀φ ∈ Inn(''G'')： φ(''H'') ≤ ''H''}}

Since {{math|Inn(''G'') ⊆ Aut(''G'')}} and a characteristic subgroup is invariant under all automorphisms, every characteristic subgroup is normal. However, not every normal subgroup is characteristic.  Here are several examples:
* Let {{math|''H''}} be a nontrivial group, and let {{math|''G''}} be the [[direct product of groups|direct product]], {{math|''H'' × ''H''}}.  Then the subgroups, {{math|{1} × ''H''}} and {{math|''H''  × {1{{)}}}}, are both normal, but neither is characteristic.  In particular, neither of these subgroups is invariant under the automorphism, {{math|(''x'', ''y'') → (''y'', ''x'')}}, that switches the two factors.
* For a concrete example of this, let {{math|''V''}} be the [[Klein four-group]] (which is [[group isomorphism|isomorphic]] to the direct product, <math>\mathbb{Z}_2 \times \mathbb{Z}_2</math>).  Since this group is [[abelian group|abelian]], every subgroup is normal; but every permutation of the 3 non-identity elements is an automorphism of {{math|''V''}}, so the 3 subgroups of order 2 are not characteristic.  Here {{math|V {{=}} {''e'', ''a'', ''b'', ''ab''} }}. Consider {{math|H {{=}} {''e'', ''a''{{)}}}} and consider the automorphism, {{math|T(''e'') {{=}} ''e'', T(''a'') {{=}} ''b'', T(''b'') {{=}} ''a'', T(''ab'') {{=}} ''ab''}}; then {{math|T(''H'')}} is not contained in {{math|''H''}}.
* In the [[quaternion group]] of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic.  However, the subgroup, {{math|{1, −1{{)}}}}, is characteristic, since it is the only subgroup of order 2.
* If {{math|''n''}} > 2 is even, the [[dihedral group]] of order {{math|2''n''}} has 3 subgroups of [[index of a subgroup|index]] 2, all of which are normal.  One of these is the cyclic subgroup, which is characteristic.  The other two subgroups are dihedral; these are permuted by an [[outer automorphism group|outer automorphism]] of the parent group, and are therefore not characteristic.

=== Strictly characteristic subgroup{{anchor|Strictly invariant subgroup}} ===
A ''{{vanchor|strictly characteristic subgroup}}'', or a ''{{vanchor|distinguished subgroup}}'', is one which is invariant under [[surjective]] [[endomorphism]]s. For [[finite group]]s, surjectivity of an endomorphism implies injectivity, so a surjective endomorphism is an automorphism; thus being ''strictly characteristic'' is equivalent to ''characteristic''. This is not the case anymore for infinite groups.

=== Fully characteristic subgroup{{anchor|Fully invariant subgroup}} ===
For an even stronger constraint, a ''fully characteristic subgroup'' (also, ''fully invariant subgroup'') of a group ''G'', is a subgroup ''H'' ≤ ''G'' that is invariant under every [[group homomorphism#Types|endomorphism]] of {{math|''G''}} (and not just every automorphism):
:{{math|∀φ ∈ End(''G'')： φ(''H'') ≤ ''H''}}.

Every group has itself (the improper subgroup) and the trivial subgroup as two of its fully characteristic subgroups. The [[commutator subgroup]] of a group is always a fully characteristic subgroup.<ref>
{{cite book
 | title = Group Theory
 | first = W.R. | last = Scott
 | pages = 45–46
 | publisher = Dover
 | year = 1987
 | isbn = 0-486-65377-3
}}</ref><ref>
{{cite book
 | title = Combinatorial Group Theory
 | first1 = Wilhelm | last1 = Magnus
 | first2 = Abraham | last2 = Karrass
 | first3 = Donald  | last3 = Solitar
 | publisher = Dover
 | year = 2004
 | pages = 74–85
 | isbn = 0-486-43830-9
}}</ref>

Every endomorphism of {{math|''G''}} induces an endomorphism of {{math|''G/H''}}, which yields a map {{math|End(''G'') → End(''G''/''H'')}}.

=== Verbal subgroup ===
An even stronger constraint is [[verbal subgroup]], which is the image of a fully invariant subgroup of a [[free group]] under a homomorphism. More generally, any [[verbal subgroup]] is always fully characteristic. For any [[reduced free group]], and, in particular, for any [[free group]], the converse also holds: every fully characteristic subgroup is verbal.